Problem 52 - Chemistry An Atoms First Approach

Question

It takes \(585 \mathrm{J}\) of energy to raise the temperature of \(125.6 \mathrm{g}\) mercury from \(20.0^{\circ} \mathrm{C}\) to \(53.5^{\circ} \mathrm{C}\). Calculate the specific heat capacity and the molar heat capacity of mercury.

Step-by-Step Solution

Answer

The specific heat capacity of mercury is \(0.139 \frac{J}{g \cdot °C}\), and the molar heat capacity of mercury is \(27.88 \frac{J}{mol \cdot °C}\).

Step 1: List the given information

We know the following: - Energy (Q) = 585 J - Mass of mercury (m) = 125.6 g - Initial temperature (T1) = 20.0°C - Final temperature (T2) = 53.5°C

Step 2: Calculate the temperature change

We need to find the change in temperature (ΔT) by subtracting the initial temperature (T1) from the final temperature (T2): ΔT = T2 - T1 ΔT = 53.5°C - 20.0°C ΔT = 33.5°C

Step 3: Use the heat capacity formula to find the specific heat capacity

We have to find the specific heat capacity(c) using the formula: Q = mcΔT 585 J = (125.6 g)(c)(33.5°C) Rearrange the formula to solve for c: c = Q / (mΔT) c = 585 J / (125.6 g * 33.5°C) c = 0.139 J/(g•°C)

Step 4: Calculate the molar heat capacity

Now, we need to find the molar heat capacity using the specific heat capacity and the molar mass of mercury. The molar mass of mercury is 200.6 g/mol. Molar heat capacity = Specific heat capacity * Molar mass of mercury Molar heat capacity = 0.139 J/(g•°C) * 200.6 g/mol Molar heat capacity = 27.88 J/(mol•°C) The specific heat capacity of mercury is 0.139 J/(g•°C), and the molar heat capacity of mercury is 27.88 J/(mol•°C).

Key Concepts

Molar Heat Capacity

Molar heat capacity refers to the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius. In the context of our exercise, we calculated the molar heat capacity of mercury using its specific heat capacity and its molar mass. Here's how you can think about it:

• Specific heat capacity is the heat required per gram. When multiplied by the molar mass, you switch from a gram basis to a mole basis.
• This conversion is crucial because molar heat capacity allows us to compare heat requirements across different substances, based on the amount of substance rather than just mass.
• The formula used was: \[\text{Molar Heat Capacity} = \text{Specific Heat Capacity} \times \text{Molar Mass of Mercury}\]
• For mercury, this gives us a molar heat capacity of 27.88 J/(mol·°C), showing how much heat is required to increase the temperature of 1 mol of mercury by 1°C.

Understanding these concepts helps in various applications, such as when we are dealing with chemical reactions or when we need to manage heat in physical systems.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat and temperature, and their relation to energy and work. At its core, thermodynamics helps us understand how energy is transferred and transformed.
When considering the heating of mercury in our exercise, thermodynamic principles tell us how much energy (in Joules) is needed for a specific temperature change.

• A key formula in thermodynamics is the heat capacity equation used: \[ Q = mc\Delta T \]Where:
  • \( Q \) is the heat energy supplied (585 J in the problem).
  • \( m \) is the mass of the substance (125.6 g here).
  • \( c \) is the specific heat capacity.
  • \( \Delta T \) is the change in temperature (33.5°C).
• This formula ties together how much heat energy is needed based on the mass and the substance's specific heat capacity.
• Understanding thermodynamics helps predict how systems will respond to changes in temperature, and it provides insights into energy efficiency and energy conversion.

Mercury Calculations

Mercury is commonly used in thermometers due to its wide liquid range and predictable expansion upon heating. Calculating the heat capacity of mercury is a great way to understand its thermal properties.

• The specific heat capacity, found using \[ c = \frac{Q}{m\Delta T} \]was calculated as 0.139 J/(g·°C), a relatively low value. This means mercury doesn't require much heat to change its temperature, contrasting with substances like water, which have higher specific heat capacities.
• With the low specific heat capacity, mercury can rapidly change temperature, which is why it's perfect in measuring devices.
• To calculate the temperature change mercury undergoes due to added heat, understanding these calculations and properties is fundamental.
• Performing the mercury calculations provides insight into its use in technology and how it's been a reliable choice historically for specific applications due to its predictable thermodynamic behavior.

Knowing mercury's specific thermal properties gives insight into not only experimental procedures but also practical everyday tool usage.